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Norton, Nicholas Charles
Languages: English
Types: Doctoral thesis
Subjects: QA

Classified by OpenAIRE into

arxiv: Mathematics::Commutative Algebra
In this thesis, we consider several generalizations of the theory of Quasi-Frobenius rings, and construct examples of the classes of rings we introduce. In Chapter 1 we establish well known results, although the way in which we use idempotents is apparently new.\ud \ud Chapter 2 is devoted to the study of three generalizations of Quasi-Frobenius rings, namely D-rings, RD-rings (restricted D-rings) and PD-rings (partial D-rings). PD-rings is the largest class of rings we study, and we show that these rings can be considered as a natural generalization of Nakayama's definition of a Quasi-Frobenius ring. D-rings are defined by annihilator conditions, and RD-rings are a generalization of D-rings. We show that RD-rings, hence also D-rings, are semi-perfect, and it follows that they are also PD-rings. We will show that in the self-injective case, these three classes of rings all coincide with a class of rings studied by Osofsky, [30]. We will investigate when the properties described are Morita invariant, and will show that finitely generated modules over D-rings are finite dimensional. Finally, we study group rings over D-rings, RD-rings and PD-rings, and in particular show that if a group ring is a D-ring, then the group is finite and the ring is a D-ring, and further, either the ring is self-injective or the group is Hamiltonian. In Chapter 3 we construct examples of D-rings, RD-rings and PD-rings.\ud \ud Chapter 4 contains results obtained jointly by Dr. C. R. Hajarnavis and the author. Here, we generalize a result of Hajarnavis, [14]., by considering Noetherian rings each of whose proper homomorphic imaeges are i.p.r.i.-rings. He will obtain a partial structure theory for such rings, and in the prime bounded case show that such rings are Dedekind prime rings.
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    • Osaka Nath. J. 1 (1949), pp.52 - 61. ASANO, K.: 'Zur Arithmetik in ·Schiefringen, II',
    • I (1950), pp. 1 - 27. BASS, H.: 'Finitistic Dimension and a Homological
    • Hath. Soc. 95 (1960), pp. 466 - 488. BJOHK, J. -E.: 'langs satisfying certain chain
    • conditions', J. fur Reine und Ange\1. l-fath.
    • 245 (1970), pp. 63 - 73. CONHELL, 1. G.:'On the group rine', Canad. J. Haths.
    • 15 (1963), pp. 650 - 685. CURTIS, C. l/. and RBIlIEn, 1.: 'Hepresentution 'l'heory of
    • Interscience Publications, 1962. DESH:e1UWE, N. G.:' Structure of riCht subdirectly
    • irreducible rings. I.', J. Algebra 17 (1971)~
    • pp. 317 - 325. DHAN,. H.: Thesis, Uni v. of Leeds, 1967. [11] GOLDIE, A. H.: 'Rings .\'lith Haximum Condition',
    • Multigraphed notes, Yale University, 1961. [12J GOLDIE, A. \'i.:'Non-commutative Principal Ideal Rings',
    • Arch. i·lath. 13 (1962), pp. 213 - 221. [13J GOLDIE, A. \1.:' Lectures on Rings and Hodules', Lecture
    • notes in Maths.,246, Springe,.r-Verlag, 1973,
    • pp. 213 - 321. [14J HAJARNAVIS, C.R.: 'Non-commutative rings 'olhose
    • London Math. Soc. 5 (1973), pp. 70 - 74. [15J HAJARNAVIS, C. R. and LENAGAN, T. H.: 'Localisation
    • in Asano Orders', J. Algebra 21 (1972), pp.441 - 449. [16J JLlI.NNULA, T. A.: 'On the construction of Quasi-Frobenius
    • rings', J. Algebra 25 (1973), pp. 403 - 414. [17J HEllSTEIN, I. N.:'Non-commutative Rings', Carus I·lath.
    • Honographs, 15, Amer. :·:ath. Soc. Publications, 1968. [18J IKEDA, M.:' Some Generalisations of. Quasi-Frobenius Rings',
    • Osaka Hath. J. 3 (1951), pp. 228 - 239. LENAGAN, T. H.:' Bounded Asano orders are hereditary',
    • Bull. London Math. Soc. 3 (1971), pp. 67 - 69. [3Lf} ROBSON, J. C.: 'Idealisers and Hereditary Noetherian Prime
    • Rings', J. AlGebra 22 (1972), pp. 45 - 81.
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